Proves limit curve theorem for incomplete metric spaces, applies to null distance in Lorentzian manifolds.
problem Control of Lorentzian lengths of limit curves in incomplete metric spaces.
method Proves limit curve theorem for incomplete metric spaces and applies to null distance.
result Strong control on Lorentzian lengths of limit curves in Sormani and Vegas' null distance.
Study shows superdiffusive behavior in geodesic flows on curved surfaces.
problem Understanding the statistical behavior of geodesic flows on curved surfaces.
method Proved nonstandard central limit theorem with superdiffusive normalisation (tlogt)1/2 for geodesic flows on nonpositively curved surfaces. result Geodesic flows exhibit superdiffusive behavior with correlations decaying at rate t−1. New proof for stable reduction theorem using Kähler-Einstein metrics.
problem Proving the stable reduction theorem for curves over punctured curves.
method Using Kähler-Einstein metrics on fibers to obtain limiting stable curves.
result A new analytic proof of the stable reduction theorem for curves over punctured curves.
Method upgrades limit theorems to mixing limit theorems for dynamical systems.
problem Improving limit theorems for dynamical systems.
method General method for upgrading limit theorems to mixing limit theorems.
result Mixing limit theorems for specific subbundles of the Kontsevich-Zorich cocycle.
Polynomial decay of correlations shown for curved surfaces.
problem Analyzing geodesic flows on curved surfaces.
method Proving polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
result Polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
Survey on random walks on mapping class groups and their properties.
problem Understanding random walks on mapping class groups.
method Analyzing actions on Teichmüller spaces and curve complexes.
result Laws of large numbers and central limit theorems for random walks.
The curve shortening flow transforms figure-eight curves into bowties.
problem Transforming figure-eight curves into a specific shape under curve shortening flow.
method Applied curve shortening flow to figure-eight curves with specific properties, proving convergence to a quadrilateral.
result The renormalized limit of the flow converges to a quadrilateral called a bowtie.
The paper calculates curvature limits and proves Gauss-Bonnet theorems in affine and Minkowski groups.
problem Computing curvature limits in affine and Minkowski groups.
method Analyzing Euclidean C2-smooth surfaces and curves in affine and Minkowski groups. result Gauss-Bonnet theorems in affine and Minkowski groups are proven.
Smooth compactness theorem for elasticae, except straight segments.
problem Compactness of elasticae space.
method Smooth compactness theorem proof.
result Smooth stability results for minimizers.
The paper calculates curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
problem Computing curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
method Sub-Riemannian limits of Gaussian curvature, Schouten-Van Kampen affine connections, and adapted connections.
result Gauss-Bonnet theorems associated with Schouten-Van Kampen affine connections in the Heisenberg group.
Compactness theorem for quasiregular curves proves normality and resolves nodal points.
problem Proving compactness and normality for quasiregular curves.
method Using Gromov's compactness theorem and bubble trees to associate a limit curve and measure.
result A nodal resolution of quasiregular curves via bubble trees.
We construct Colding-Minicozzi limit minimal laminations in open domains in $\rth$ with the singular set of C1-convergence being any properly embedded C1,1-curve. By Meeks' C1,1-regularity theorem, the singular set of convergence of a Colding-Minicozzi limit minimal lamination L is a locally finit…
We extend the notion of a pseudoholomorphic vector of Iwaniec, Verchota, and Vogel to mappings between Riemannian manifolds. Since this class of mappings contains both quasiregular mappings and (pseudo)holomorphic curves, we call them quasiregular curves. Let n≤m and let M be an oriented Riemannian n-manifold,…
We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in C2 with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.
The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.
problem Understanding exceptional sets for radial limits of superharmonic functions on curved manifolds.
method Analysis of radial geodesic rays, Poisson integrals, Green potentials, and Riesz decomposition.
result Sharp bounds on Hausdorff dimensions of exceptional sets for superharmonic functions.
A simplicial complex is called negatively curved if all its simplices are isometric to simplices in hyperbolic space, and it satisfies Gromov's Link Condition. We prove that, subject to certain conditions, a compact graph of spaces whose vertex spaces are negatively curved 2-complexes, and whose edge spaces are points …
Let M be a closed Riemannian manifold with a parallel 1-form Ω. We prove two theorems about the curve shortening flow in M. One is that the {\csf} $\ct$ in M exists for all t in [0,∞), if it satisfies Ω(T)≥0 on the initial curve $\co$. Here T is the unit tangent vector on $\co$. The other one …
Develops thermodynamic formalism for quasimorphisms on negatively curved spaces.
problem Analyzing quasimorphisms on negatively curved spaces.
method Thermodynamic formalism framework, Banach isomorphism, weak Livšic cohomology.
result Establishes Central Limit Theorem and invariance principle for unbounded quasimorphisms.
We give explicit bounds on the intersection number between any curve on a tight multigeodesic and the two ending curves. We use this to construct all tight multigeodesics and so conclude that distances in the curve graph are computable. The algorithm applies to all surfaces. We recover the finiteness result of Masur-Mi…
Paper extends Schur's theorem to spherical curves via monotonicity.
problem Comparing chord lengths of convex and spherical curves.
method Monotonicity and expansion module approach.
result Schur's Theorem extended to spherical curves.
New theorem counts curves on orbifolds.
problem Counting curves on surfaces.
method Applied Mirzakhani's theorem to orbifolds.
result Curve counting theorem extends to orbifolds.
The study proves unique and isolated properties of Einstein 5-manifolds via gap theorems in 4 dimensions.
problem Understanding the structure and regularity of Einstein 5-manifolds.
method Analysis of tangent cones, gap theorems for 4-dimensional orbifolds, and careful metric analysis.
result Noncollapsed limits of Einstein 5-manifolds have unique and isolated tangent cones.
The paper proves a generalized inverse function theorem for curved L∞ spaces.
problem Proving a generalized inverse function theorem for curved L∞ spaces. method Obstruction theory for L∞ homomorphisms and homotopy transfer theorem for curved L∞ algebras. result A morphism of curved L∞ spaces which is a quasi-isomorphism at a point has a local homotopy inverse. Defines new metrics for Lorentzian spaces and their convergence.
problem Defining metrics for Lorentzian spaces and their convergence.
method Abstract approach to Lorentzian Gromov-Hausdorff distance and convergence, defining bounded Lorentzian-metric spaces, and proving stability under GH limits.
result GH limits of Lorentzian-metric spaces are isometric and homeomorphic.
We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on Out(FN), each time under a finite second moment condition on the measure (either with respect to the Teichmüller metric, or with respect to the Lipschitz metric …
New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.
problem Rigidity of sharp spectral gap in nonnegatively curved spaces.
method Mixing Sobolev theory and singular 1D-localization.
result Rigidity of λ=diam2π2 in compact RCD(0,N) spaces. Quantifies Schur's theorem for curves in CAT(k) spaces.
problem Quantifying Schur's comparison theorem for curves in CAT(k) spaces.
method Comparison formula for curves in model planes, curvature measures, moment arm, and Reshetnyak's theorem.
result Sharpens and extends classical arm and bow lemmas and Riemannian analogues.
The study extends removability results for quasiregular curves in Euclidean spaces.
problem Removability of singularities in quasiregular curves.
method Extending a fundamental inequality for volume forms to calibrations and proving a Caccioppoli inequality for quasiregular curves.
result Every non-constant quasiregular curve has infinite energy.
Defines signed quasiregular curves and proves growth theorem.
problem Understanding growth of signed quasiregular curves.
method Proves weak reverse Hölder inequality and uses it to prove growth theorem.
result Proves growth theorem for signed quasiregular curves.
We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of s…
Paper extends theorem on covering spaces and Jordan curves.
problem Covering and extending theorems for Alexandrov spaces.
method Introduces proximal homotopic cycles to extend the Mitsuishi-Yamaguchi theorem.
result Extensions of the Mitsuishi-Yamaguchi Good Covering Theorem and Jordan curve theorem.
Theorem converse to Jordan's curve theorem says that {\it if a compact set K has two complementary domains in R2, from each of which it is at every point accessible, it is a simple closed curve}. We show that the requirement of this theorem that {\it all} points of K were accessible from {\it both} complementa…
In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds (M_i,g_i), i \in N, whose Ricci curvature is not less than -c^2(i), where c^2(i…
Multivariate Poisson approximation of the length spectrum of random surfaces is studied by means of the Chen-Stein method. This approach delivers simple and explicit error bounds in Poisson limit theorems. They are used to prove that Poisson approximation applies to curves of length up to order o(loglogg) with g…
The paper studies Kähler-Einstein metrics with singularities and their limits.
problem Analyzing Kähler-Einstein metrics with crossing edge singularities and their limits.
method Extending Guenancia's techniques, the paper shows convergence of metrics under specific angle conditions.
result Negatively curved Kähler-Einstein crossing edge metrics converge to mixed cusp and edge metrics smoothly away from the divisor.
Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, we prove codimension bounds for ancient mean curvature flows by their tangent flow at −∞, generalizing a theorem for cylinders in [CM19b]. In the case of the m-covered circle, we apply this bound to prove a stron…
The study proves a discrete version of Segre's theorem for polygonal curves.
problem Proving a discrete analog of a four-vertex theorem for spherical curves.
method Using the concept of discrete tangent indicatrix of a polygon.
result A polygon with at least four vertices and a non-self-intersecting discrete tangent indicatrix has at least four flattenings.
Study magnetic curves in Sasakian manifolds, classifying and parametrizing them.
problem Classify and parameterize pseudo-Hermitian magnetic curves in Sasakian manifolds.
method Define and classify pseudo-Hermitian magnetic curves, construct parametrizations.
result Complete classification theorem for pseudo-Hermitian magnetic curves in Sasakian manifolds.
Generalizes Toponogov theorem to Alexandrov spaces.
problem Estimating curve length in non-Euclidean spaces.
method Generalization of Toponogov theorem.
result Proved the length of a curve in two-dimensional Alexandrov spaces.
Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result f…
Loewner's theorem connects two curve properties via simple functions.
problem Understanding the rotation number of curves defined by smooth periodic functions.
method Elementary proof following Bol's work.
result Two curve properties have non-negative rotation number.
We verify if Gausssian curvature of surfaces and normal curvature of curves in surfaces introduced by Diniz-Veloso arXiv:1210.7110 and by Balogh-Tyson-Vecchi arXiv:1604.00180 to prove Gauss-Bonnet theorems in Heisenberg space H1 are equal. The authors in arXiv:1604.00180 utilize a limit of Gaussian and norma…
Condition for embedding metric spaces into curved manifolds.
problem Embedding conditions for metric spaces in curved manifolds.
method If-and-only-if condition on five-point metric spaces.
result Five-point metric spaces admit embeddings into nonnegatively curved Riemannian manifolds.
The Tait-Kneser theorem states that the osculating circles of a plane curve with monotonic curvature are pairwise disjoint and nested. We discuss this theorem and a number of its variations.
The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations…
We establish a Lehto--Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space, which are useful for establishing a big Picard-type theorem and a big Brody-type one for holomorphic curves.
The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.
problem Finding bounds on intersections and inflections for spherical polygons.
method Adapting smooth curve theorems to spherical polygons using discrete tools.
result Proves discrete analogs of four-vertex theorems for spherical polygons.
Study of curves in dual space with constant curvature and torsion.
problem Classifying curves in dual space with specific geometric properties.
method Defined curvature and torsion for curves in dual space, classified curves with constant properties, and proved existence theorems.
result Established fundamental theorem of existence for dual curves with prescribed curvature and torsion.